 In this video I'm going to be talking about how to solve linear systems in two variables. So a system of equations is simply just a set of equations where you have multiple equations and multiple variables. That's basically what a system is. So in this case I'm just going to be solving one using two variables. I'm just going to do two examples of substitution and then in the next video I'm going to do two examples of elimination. So when you're using substitution to solve a system you have to have a single variable by itself. So notice we have a y over here that's all by itself. So this system is set up where we want it to. It has a variable by itself and we can use that to basically plug into the second one. So I'm going to use the fact that y is equal to x minus 1 and I'm actually going to plug that in to the second equation for y. Since y is equal to this I'm going to plug that in for y. So this is what it's going to look like, x plus x minus 1 equals 7. So notice it took out the y and replaced it with an x minus 1. That right there is substitution, that's kind of blatantly obvious, that's what substitution is. Taking something else out and putting something else in that's equivalent. Now what you do is now you have a single variable equation and you just simply need to solve that equation. Now I put parentheses here but I don't necessarily need them. I'm not multiplying by anything out front so there's really no need for them. But I like to have the parentheses there to emphasize the point that I substituted. But anyway, combine some like terms on this side, so I get 2x minus 1 is equal to 7 and then I'm going to add 1 to both sides, add 1 to both sides and then divide everything by 2, divide everything by 2, so I get that x is equal to 4. Now what I've done is I've solved for a single variable. Now what I'm going to do is I'm going to take this and I'm going to plug it back into my first equation. I'm going to plug it back into the equation that I kind of used for substitution. In that case y is equal to x minus 1, y is equal to 4 minus 1, y is equal to 3. So I can actually very quickly solve for what the other variable is going to be. My solution, there's a number of different ways to write the solution. I usually write it as coordinates 4, 3 because we're using x's and y's, that's what I usually write for the solution. That basically gives me a point where these two lines intersect, that's one way to look at it. Or you can simply write that x is equal to 4 and that y is equal to 3. That's another way to kind of write out what your solution is. That's a very quick, very easy example of how to use substitution. I'm going to do one more example using substitution. The one thing that you notice first is that when you're looking at this problem we don't have a variable that is solved or that is by itself. That's one of the first things I have to do is I have to take one of my equations and solve for a single variable. You can take whatever variable you want to, but you want to try to make this as easy for yourself as possible. Notice here that x right here is by itself, there's no number in front of it. That one right there is going to be the easiest one to solve for because all I have to do is take this 2y and subtract it to the other side over there with 4. x is equal to negative 2y plus 4. Notice the difference, take this 2y, since it's positive I subtracted it over to the other side. What I have here is I have a variable by itself. I have something to substitute in with. I'm going to take this portion of the equation since that's what x is equal to and I'm going to plug that into the second equation in for x. That's going to give me 3 times negative 2y plus 4 minus 4y equals 7. Notice the x came out and we plugged in negative 2y plus 4. Just like last time I have a single equation with a single variable. I'm just going to solve for that single variable. A couple of things I have to do first, I have to take this 3 times everything. I have to distribute first negative 6y plus 12 minus 4y equals 7. Now I'm just going to do some combining like terms. I'm going to subtract negative 6 and negative 4 is going to make a negative 10y plus 12 equals 7. This 12, I need to get the numbers on the other side so I have to subtract 12 over to the other side. Negative 10y equals 7 minus 12 is going to be a negative 5. Then after that divide by a negative 10y is going to be, divide by a negative 10 here that's going to reduce to a 1 half. A positive 1 half because it's going to be negative divided by a negative. That is what y is equal to and just like last time we're going to take this and plug this back in to what we used for substitution to figure out what x is going to be. x equals negative 2y plus 4. What I'm going to do is take out the y and plug in a 1 half. x is equal to negative 2 times 1 half plus 4. Negative 2 times 1 half is simply just going to be negative 1 plus 4. Therefore x is equal to 3. My solution how I usually write this as a coordinate, got to get the parentheses on your coordinates. There we are. My solution is 3 comma 1 half, 3 comma 1 half. My x coordinate is 3 and my y coordinate is 1 half. Alright that's just a couple of examples of using substitution. In the next video I will do a couple of examples of solving using elimination.